Efficient Computation of Distance Functions for Navigation Vector Fields in Lie Groups
For robotics applications requiring real-time control on embedded platforms, this method speeds up distance computation for vector-field-based path tracking in Lie groups.
The paper proposes a method to efficiently compute the distance between a point and a G-polynomial curve in Lie groups, reducing computation time significantly while maintaining accuracy compared to optimization-based approaches. The method is validated on a robotic manipulator.
Vector-field-based methods are widely used for robot control and are often applied to the path-tracking problem. Some vector field approaches require repeatedly computing the distance between the robot configuration and the curve, as well as the corresponding closest point. Recently, vector fields have been extended to Lie Groups. In this case, this computation can be expensive, especially when performed at high control frequencies on embedded platforms. This paper proposes a method for efficiently computing the distance between a point and a curve represented as what is called a G-polynomial curve, which is a curve representation that generalizes polynomial curves to matrix Lie groups. The proposed approach exploits the structure of these curves to reduce the problem to a small number of polynomial root-finding computations. Simulation results show that the method significantly reduces computation time while maintaining accuracy compared to existing optimization-based approaches. Practical formulas are also provided for the case of the group SE(3), and the method is validated experimentally on a robotic manipulator. The methodology is implemented in a computational package, available online.